Let $F\in L^2(\Omega)$ be such that $-\text{div}F\in W^{-1,2}(\Omega)$ (Dual of the Sobolev Space $W_0^{1,2}(\Omega)$) be non-negative where $\Omega$ is a bounded domain in $\mathbb{R}^N$.

My question is about the existence of an approximation of $-\text{div}F$ in such a way that

(1) there exists a sequence of function $F_n\in (W^{1,\infty}(\Omega))^N$ which converges to $F$ in $L^2(\Omega)$ with the property $F_n\leq F$ in $\Omega$

(2) Moreover, $-\text{div}(F_n)\geq f$ for some $f\geq 0$ in $\Omega$ and $f\geq c_{k}>0$ for all $k\subset\subset\Omega$.

Can you kindly help me whether such approximation is possible or not, even if one imposes some extra hypothesis on $F$?

Thanks in advance.


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